I didn't have the facility here to listen, but; buzz near the nut, just on the G-string, disappears once fretted? Sounds to me like the nut slot for your G-string has been filed too low.
A temporary check would be to loosen the string off, place a sliver of matchstick wood in there and tune up and see if it's gone. If that is your cause, your options depend on how much you want to spend. You might be able to get away with dropping in some hard-setting glue. If the string gauge is lighter than you like, a move from, say, 40 to 45 might have it sitting higher.
The best solution is to have a tech do a good setup, part of which would involve putting in a fresh, correctly cut nut. Something around €40-50, which seems like a big chunk of the cost of an OLP but they're a nice bass and a properly set up one means more fun playing. The correct files for doing nuts are expensive, and it's easy to get it wrong, so I recommend asking a tech.
You could look at John Carruthers' excellent 4-part video on how to set up a bass on Youtube, do the other jobs and, if your frets etc are in good shape, just ask for the nut replacement to see if you're quoted a reduced price.